Regular Polygonal Partitions of a Tverberg Type

Abstract

A seminal theorem of Tverberg states that any set of $$T(r,d)=(r-1)(d+1)+1$$ points in $${\mathbb {R}}^{d}$$ can be partitioned into r subsets whose convex hulls have non-empty r-fold intersection. Almost any collection of fewer points in $${\mathbb {R}}^{d}$$ cannot be so divided, and in these cases we ask if the set can nonetheless be P(r, d)-partitioned, i.e., split into r subsets so that there exist r points, one from each resulting convex hull, which form the vertex set of a prescribed convex d-polytope P(r, d). Our main theorem shows that this is the case for any generic $$T(r,2)-2$$ points in the plane and any $$r\ge 3$$ when $$P(r,2)=P_{r}$$ is a regular r-gon, and moreover that $$T(r,2)-2$$ is tight. For higher dimensional polytopes and $$r=r_{1}\cdots r_{k}$$ , $$r_{i} \ge 3$$ , this generalizes to $$T(r,2k)-2k$$ generic points in $${\mathbb {R}}^{2k}$$ and orthogonal products $$P(r,2k)=P_{r_{1}}\times \cdots \times P_{r_{k}}$$ of regular polygons, and likewise to $$T(2r,2k+1)-(2k+1)$$ points in $${\mathbb {R}}^{2k+1}$$ and the product polytopes $$P(2r,2k+1)=P_{r_{1}}\times \cdots \times P_{r_{k}}\times P_{2}$$ . As with Tverberg’s original theorem, our results admit topological generalizations when r is a prime power, and, using the “constraint method” of Blagojević, Frick, and Ziegler, allow for dimensionally restricted versions of a van Kampen–Flores type and colored analogues in the fashion of Soberón.